Calculus
4.1--4.2 Review Notes
4.1
o
Name:________________________________________
Name:________________________________________
Mitchell
You do not need to use u substitution if the expression within the integral is written, or can be written, so that
each term is in the form ax b (where a and b are rational numbers).
EXAMPLES:
 3x 8 − x 3 + 7 
dx
∫ 
x5

∫ (4x − 1) dx
2
o
2x 3 + x 2 − 12x + 9
dx
∫
x+3
2
 3

∫  z 5 − z 3 + 6z  dz
∫ (5x − 3)(2x + 5)dx
You do not need to use u substitution if the expression within the integral is the derivative of sine, cosine,
tangent, secant, cosecant, or cotangent.
EXAMPLES:
o
∫ sinudu = − cos u + C
∫ cos udu = sinu + C
∫ sec
2
udu = tanu + C
∫ sec u tanudu = sec u + C
∫ csc ucot udu = − csc u + C
∫ csc
2
udu = − cot u + C
If the expression in the integral looks similar to the trig integrals above except for the “inside” , then you can set
the “inside” of the trig function(s) equal to u.
EXAMPLES:
∫ 4 cos(
o
1
2
x)dx
∫ cos 4xdx
2
∫ csc(cos 5x)cot(cos 5x)sin5xdx
5x
∫
x2 − 3
dx
sin(6x)
∫ cos (6x) dx
5
∫ cos (5t)sin(5t)dt
3
x2
x
3
) tan( 3x )dx
dx
∫ (2x
3
+ 1)7 x 2 dx
∫
5x + 7dx
∫ (1 − sin (3t))cos(3t)dt
2
∫
sin(2x)
dx
1 − cos(2x)
∫ cos (5t)sin(5t)dt
3
If there is a tangent to any power and secant squared, set the tangent equal to u (because the derivative of
tangent is secant squared). If there
there is a cotangent to any power and cosecant squared, set the cotangent equal to
u (because the derivative of cotangent is cosecant squared).
EXAMPLES:
o
3
∫ sec(
If the integral has a sine and cosine, then the one is raised to an exponent should be set equal to u.
EXAMPLES:
o
∫
cos 3 x
2
If there is a polynomial expression raised to an exponent (other than 2) or the root of a polynomial expression,
then you can set the polynomial equal to u.
EXAMPLES:
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∫ x cot(x )csc(x )dx
∫ cot
3
(7x)csc 2 (7x)dx
If the methods above do not initially work, try rewriting the expression so that one of these methods does work.
EXAMPLES:
5
∫ tan 4x sin 4x dx
∫ sin(2x)sec (2x)dx
5
csc(2x)
∫ sin(2x) dx
1
∫ sin (5x) dx
2